Semidiscrete Ritz-Galerkin approximation of nonlinear parabolic boundary control problems -- strong convergence of optimal controls (Q1322719)

The paper discusses an optimal control problem for the linear parabolic equation \(y_ t= \Delta y-y\) with nonlinear Neumann type boundary conditions \(\partial y/\partial n= b(t,x,y,u)\). Controls \(u\) belong to a standard convex set. Additional constraints are of the form of inequalities for integrals over the spatial domain. Under some regularity assumptions and the assumption that the second derivative of the Lagrange function with respect to the state and controls is strongly positive at the local minimum \((y_ 0, u_ 0)\), the author proves that the solutions of Ritz-Galerkin semidiscrete approximations strongly converges to \((y_ 0, u_ 0)\).